1. Field of the Invention
The present invention is directed to an image reconstruction method of the type suitable for use for producing a displayed image from cone beam data or pyramidal beam data in an x-ray computed tomography apparatus.
2. Description of the Prior Art
Algorithms are known in computer tomography (CT) for the exact reconstruction of tomographic cone beam data or pyramidal beam data wherein radon values in the 3D radon space are first calculated by line integration over the two-dimensional detector (Mathematical Methods in Tomography, G. T. Herman, A. K. Louis, F. Natterer, Eds., Lecture Notes in Mathematics, Springer Verlag 1991). FIG. 1 schematically shows the focus 1 of an X-ray source that emits a pyramidal X-ray beam 2 that strikes a two-dimensional detector 3. The detector 3 is formed by a matrix of detector elements whose output signals are supplied to a computer 4 that calculates an image of the examination subject 5 therefrom and effects the playback on a monitor 6. The examination subject 5 is transirradiated from various projections. To this end, the X-ray source and the detector 3 and, thus, the X-ray beam 2 as well are rotated around the examination subject 5 around the system axis 7. The radial derivative of the radon transform is determined by integration along lines, for example the line 8, over data from detector elements lying in those lines. The identified radon value, or the identified radial derivative of the radon value, corresponds to the surface integral over the integration plane 9. On the basis of this procedure, the 3D radon values on a spherical coordinate system r, .crclbar., .phi. in the 3D radon space are first determined from a number of projections. As described by Marr et al. in Proc. Mathematical Aspects of Computerized Tomography, Oberwolfach (FRG), 1980, G. T. Herman, F. Natterer, Eds., Springer Verlag, 1981, the subject can be reconstructed in a two-stage process proceeding from these 3D radon values. In the first step, a 2D parallel projection of the subject perpendicular to the .phi.-plane is generated on every .phi.-plane by ordinary 2D radon inversion. In the second step, the subject in this slice is reconstructed, again by ordinary 2D radon inversion on every vertical plane. This procedure is illustrated in FIG. 2. FIG. 2 shows a two-stage reconstruction algorithm according to Marr et al. First, respective 2D radon inversions are implemented on all .phi.-planes. As a result, a 2D parallel projection through the subject perpendicular to the .phi.-plane is calculated on every .phi.-plane. Each row or line of such a 2D parallel projection belongs to a different vertical (z-const.) plane. Using all values belonging to a vertical plane, the subject in this slice is reconstructed, again by 2D radon inversion on this vertical plane. This is implemented for all vertical planes and thus supplies the reconstructed subject volume 10.
Most of these exact reconstruction algorithms assume that the subject does not extend beyond the ray pyramid of the projection in any projection. This condition cannot be met in medical CT scanners. Other algorithms enable the reconstruction of a sub-volume of interest in a long subject when this sub-volume is bounded by two planes and the focus path contains two circular paths on these boundary surfaces (U.S. Pat. No. 5,463,666). Thus, for example, a sub-volume of interest can be reconstructed from a spiral scan when the spiral scan is supplemented by a respective circle at the top and bottom on the boundary surfaces of this sub-volume (see FIG. 3). FIG. 3 shows the reconstruction of a sub-volume 11 of interest from the examination subject 5, which is a long subject in this case, according to U.S. Pat. No. 5,463,666 with a spiral focus path 12 expanded by two circles. For calculating the plane integral corresponding to a 3D radon value, this plane--as shown in FIG. 4--is divided into a number of sub-regions, each of which is covered by a projection. The sub-area of interest of this integration plane is shown shaded in FIG. 4. It arises from the intersection of the integration plane with the sub-volume of interest. For the reconstruction of the sub-volume of interest, the plane integrals are only allowed to extend over the sub-area of interest. To that end, it is necessary that the limitation of the sub-area of interest is a straight line and that a focus position lies on this line. FIG. 4 shows the intersection lines 13 of the integration plane 9 with the limiting surfaces of the sub-volume 11 of interest. The sub-area 14 of interest is shown shaded. This sub-area 14 of the integration plane is covered by combining a number of projections.
As in FIG. 4, FIG. 5 shows the integration over the sub-area of interest shaded. In FIG. 5, however, no focus position is located on the upper boundary line of the sub-area of interest 14. Some of the rays required for the calculation of the integral over the sub-area 14 of interest are falsified by contributions of regions outside the sub-volume 11 of interest (as an example, a ray 15 is entered that also receives contributions on the section (shown bold) outside the sub-volume 11 of interest). The integral therefore cannot be formed over the sub-area 14 of interest.
In all known algorithms, a sub-volume of interest is defined, and a consistent 3D radon set is generated by limiting the plane integration to the sub-volume of interest.
U.S. Pat. No. 5,500,883 discloses a computer tomography apparatus with a cone beam or a pyramidal beam, whereby the radon values are calculated using a number of computers that divide the radon space among them such that the individual computers respectively handle regions of the radon space that are of approximately the same size. In detail, each computer handles a number of vertical planes in the radon space that are distributed over 180.degree. in the .phi.-direction at identical angular spacings and that respectively contain a polar coordinate system for the radon values belonging to that vertical plane. A spiral scan is not provided.